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Modern methods of algebra and number theory and its applications


Work number - P 21 FILED

Presented National Pedagogical Dragomanov University

Authors:
1. GRIGORCHUK R.I. – Doctor of Science in Physics and Mathematics, Professor, Distinguished Professor at Texas University A&M (USA);
2. ZHUCHOK A.V. – Doctor of Science in Physics and Mathematics, Professor, Honored Science and Technology Worker of Ukraine, Head of the Department of Algebra and System Analysis at Luhansk Taras Shevchenko National University;
3. ZHUCHOK Yu.V. – Doctor of Science in Physics and Mathematics, Professor, Professor of the Department of Algebra and System Analysis at Luhansk Taras Shevchenko National University;

Main Aim of Research is a development of new effective methods and improvement existing approaches to problems of general algebra and number theory. The methods have different applications to wide range of areas of modern mathematics such that number theory, theories of groups, semigroups, doppelsemigroups, associative rings, moduli, Lie algebras, Leibnitz algebras, dimonoids and trioids, fractals, hypergraphs, geometry, automata, formal languages, algorithms, cryptography, functions, probabilities. Most of the methods was produced in the way to solve series of well-known mathematical problems: Fields Medalist     D. Milnor conjecture on the growth of groups, Fields Medalist M. Atiyah conjecture on manifolds with irrational l2-Betti number, von Neumann-Day conjecture on the non-elementary amenable groups, Arnold-Krylov’s problem on generalization of Birkhoff’s ergodic theorem, Magnus-Chandler’s problem on the existence of new classes of just infinite groups, Rosenblatt conjecture on the superamenability of groups, Kegel’s problems on the sum of two nilpotent Lie algebras and on the product of three nilpotent subgroups, Freudenberg’s problem on nilpotency of Lie algebras of derivations, Kemhadze problem on non-primary groups, Jennings-Krasilnikov conjecture on metabelian radical Lie rings, Plotkin’s problem on automorphisms of the endomorphism semigroup of a free algebras, and many other important problems of theories of Lie algebras, Leibnitz algebras, modules over group rings, group factorizations, topologic and metric theories of real numbers for different numeration systems, fractal theory of real numbers and its applications. These methods and it’s applications become generators for genesis and development of new branches in the group theory – theories of selfsimilar groups, branch groups, iterated monodromy groups, infinite dimensional linear groups, in the universal algebra – theories of dimonoids, digroups, trioids, Leibnitz algebras, in the fractal analysis – topological and metric theory of real numbers for different numeration systems.

Number of publications: 16 monographs (9 published abroad), 841 papers (incl. 355 papers in English-language high-ranking journals). The total number of citations/h-index for the databases: Web of Science – 2259 / 68, Scopus – 2417 / 70, Google Scholar – 9576 /135. In the area of the research 12 Dc. Sc. and 74 PhD dissertations were defended.

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